Simplify and expand the following expression: $ \dfrac{2k - 7}{k + 1}-\dfrac{k}{5k - 4} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(k + 1)(5k - 4)$ Multiply the first term by $\dfrac{5k - 4}{5k - 4}$ $ \begin{align*} \dfrac{2k - 7}{k + 1} \times \dfrac{5k - 4}{5k - 4} & = \dfrac{(2k - 7)(5k - 4)}{(k + 1)(5k - 4)} \\ & = \dfrac{10k^2 - 43k + 28}{(k + 1)(5k - 4)}\end{align*} $ Multiply the second term by $\dfrac{k + 1}{k + 1}$ $ \begin{align*} \dfrac{k}{5k - 4} \times \dfrac{k + 1}{k + 1} & = \dfrac{(k)(k + 1)}{(5k - 4)(k + 1)} \\ & = \dfrac{k^2 + k}{(5k - 4)(k + 1)}\end{align*} $ Now we have: $ = \dfrac{10k^2 - 43k + 28}{(k + 1)(5k - 4)} - \dfrac{k^2 + k}{(5k - 4)(k + 1)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{10k^2 - 43k + 28 - (k^2 + k)}{(k + 1)(5k - 4)} $ $ = \dfrac{10k^2 - 43k + 28 - k^2 - k}{(k + 1)(5k - 4)} $ $ = \dfrac{9k^2 - 44k + 28}{(k + 1)(5k - 4)}$ Expand the denominator: $ = \dfrac{9k^2 - 44k + 28}{5k^2 + k - 4}$